Linearly independent vertices and minimum semidefinite rank

被引：34

作者：

Hackney, Philip ^{[2
]}

Harris, Benjamin ^{[3
]}

Lay, Margaret ^{[4
]}

Mitchell, Lon H. ^{[5
]}

Narayan, Sivaram K. ^{[1
]}

Pascoe, Amanda ^{[6
]}

机构：

[1] Cent Michigan Univ, Dept Math, Mt Pleasant, MI 48859 USA

[2] Purdue Univ, Dept Math, W Lafayette, IN 47907 USA

[3] Brown Univ, Dept Math, Providence, RI 02912 USA

[4] Grinnell Coll, Dept Math & Comp Sci, Grinnell, IA 50112 USA

[5] Virginia Commonwealth Univ, Dept Math, Richmond, VA 23284 USA

[6] Furman Univ, Dept Math, Greenville, SC 29613 USA

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2009年 / 431卷 / 08期

基金：

美国国家科学基金会;

关键词：

Minimum semidefinite rank; Join; Linearly independent vertices; GRAPHS; REPRESENTATIONS; MATRICES;

D O I：

10.1016/j.laa.2009.03.030

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs. We present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph, and conjecture that the resulting graph invariant is equal to minimum semidefinite rank. Rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs. We also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex. (C) 2009 Elsevier Inc. All rights reserved.

引用

页码：1105 / 1115

页数：11

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